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Noncontrollable finite-state channels (FSCs) are FSCs in which the channel inputs have no influence on the channel states, i.e., the channel states evolve freely. Since single-letter formulae for the channel capacities are rarely available for general noncontrollable FSCs, computable bounds are usually utilized to numerically bound the capacities. In this paper, we take the delayed channel state as part of the channel input and then define the directed information rate from the new channeldoi:10.1109/tit.2012.2201341 fatcat:qhlhfz3awbbkne35jeuch5gkfi